Levy area for the free Brownian motion: existence and non-existence

摘要

This note extends the work of Capitaine (J. Funct. Anal. 179 (1) (2001) 153) on the Levy area process for the free Brownian motion in two directions. First, we reprove that a Levy area for the Free Brownian motion exists in the Von Neumann tensor product, by exhibiting a non-commutative Burkholder-Davis-Gundytype inequality. Then, we show that there does not exist a Levy area in the projective tensor product. (C) 2003 Elsevier Science (USA). All rights reserved. [References: 5]